Theorem omnaddcl

Description: The naturals are closed under natural addition. (Contributed by Scott Fenton, 20-Aug-2025)

	Ref	Expression
Assertion	omnaddcl	⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ( 𝐴 +no 𝐵 ) ∈ ω )

Proof

Step	Hyp	Ref	Expression
1		nnon	⊢ ( 𝐴 ∈ ω → 𝐴 ∈ On )
2		naddoa	⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ ω ) → ( 𝐴 +no 𝐵 ) = ( 𝐴 +o 𝐵 ) )
3	1 2	sylan	⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ( 𝐴 +no 𝐵 ) = ( 𝐴 +o 𝐵 ) )
4		nnacl	⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ( 𝐴 +o 𝐵 ) ∈ ω )
5	3 4	eqeltrd	⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ( 𝐴 +no 𝐵 ) ∈ ω )